The result means that both teams are still searching for their first win of the campaign.
Frederic Michalak had three opportunities to reduce the deficit from penalty kicks, but missed twice before finally converting one.
However, Jackson sent two penalties sailing over in quick succession to reassert Ireland's authority before the half-time buzzer.
France failed to use the ball from a scrum four minutes after the restart, giving Jackson another chance to add to Ireland's total.
This time he was unable to do so. His kick was straight but did not have the distance to carry over.
He could have further eaten into the margin had his 58th-minute penalty found the mark, but he kicked well wide to hand Ireland yet another reprieve.
Quick thinking from the French front line allowed Louis Picamoles to squeeze over for a try when Parra had thought that he had scored.
Michalak was restored to kicking duties, and he sent it right down the middle to draw his country level at 13-13.
Keith Earls then thought that he had snatched a dramatic victory with an immediate strike back, but it was Picamoles who got back to deny him at the last.